An Elementary Proof of the $\widetilde O(n^{1/3})$ Bound for Separating Words

Chen Xu

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arXiv · 2609.08191

An Elementary Proof of the $\widetilde O(n^{1/3})$ Bound for Separating Words

Chen Xu

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Abstract

For two distinct binary words of length $n$, the separating words problem asks for a small deterministic finite automaton that accepts exactly one of them. Chase proved a $\widetilde O(n^{1/3})$ upper bound using a complex-analytic estimate for sparse polynomials. We replace that estimate by a finite-difference argument and a second-order real recurrence cutoff. The resulting elementary proof gives an explicit bound of $O(n^{1/3}(\log n)^{7/3})$ states.

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Chen Xu. 2026-09-08. An Elementary Proof of the $\widetilde O(n^{1/3})$ Bound for Separating Words. https://arxiv.org/abs/2609.08191

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